What is the solution for ((dy)/(dx))^2+2e^x(dy)/(dx)+e^{2x}=0 ?

The solution for ((dy)/(dx))^2+2e^x(dy)/(dx)+e^{2x}=0 is y=-e^x+c_{1}

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((dy)/(dx))^2+2e^x(dy)/(dx)+e^{2x}=0

(\frac{d y}{d x})^{2} + 2 e^{x} \frac{d y}{d x} + e^{2 x} = 0

y = - e^{x} + c_{1}

Solution steps

(\frac{d y}{d x})^{2} + 2 e^{x} \frac{d y}{d x} + e^{2 x} = 0

Substitute

\frac{d y}{d x}

with

y^{'} (x)

(y^{^{'}} (x))^{2} + 2 e^{x} y^{^{'}} (x) + e^{2 x} = 0

Isolate

y^{'} (x) : y^{'} (x) = - e^{x}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

y = \int - e^{x} d x

\int - e^{x} d x = - e^{x} + c_{1}

y = - e^{x} + c_{1}

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What is the solution for ((dy)/(dx))^2+2e^x(dy)/(dx)+e^{2x}=0 ?
The solution for ((dy)/(dx))^2+2e^x(dy)/(dx)+e^{2x}=0 is y=-e^x+c_{1}